To find the missing length \( l \) of the triangular prism when the surface area is given, we start by finding the surface area of the prism.
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Calculate the area of the triangular base: The area \( A \) of a triangle can be calculated using the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Here, the base of the triangle is \( 8 , \text{cm} \) and the height is \( 6 , \text{cm} \): \[ A = \frac{1}{2} \times 8 \times 6 = 24 , \text{cm}^2 \]
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Calculate the areas of the three rectangular faces:
- The bottom rectangular face has dimensions \( 8 , \text{cm} \times l \).
- The left rectangular face has dimensions \( 10 , \text{cm} \times l \).
- The right rectangular face also has dimensions \( 10 , \text{cm} \times l \).
Thus, the areas of the rectangular faces are:
- Bottom: \( 8l \)
- Left: \( 10l \)
- Right: \( 10l \)
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Combine the areas to find the total surface area: The total surface area \( S \) of the prism includes the area of the two triangular faces and the three rectangular faces: \[ S = 2 \times \text{(area of triangle)} + \text{(area of bottom rectangle)} + \text{(area of left rectangle)} + \text{(area of right rectangle)} \] \[ S = 2 \times 24 + 8l + 10l + 10l = 48 + 28l \]
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Set the total surface area equal to 524 square centimeters: \[ 48 + 28l = 524 \] Now, solve for \( l \): \[ 28l = 524 - 48 \] \[ 28l = 476 \] \[ l = \frac{476}{28} \] \[ l = 17 \text{ cm} \]
Thus, the missing length \( l \) of the triangular prism is 17 cm.